Question

So I did #67 and I got -8(correct) but I’m trying to figure out how people were able to get the other answer which is 6/5

Answer 1

We can divide both sides by the expression on either side:

$|3x+1|=|2x-7|\implies\dfrac{|3x+1|}{|2x-7|}=1\implies\left|\dfrac{3x+1}{2x-7}\right|=1$

Then we use the definition of absolute value:

$|x|:=\begin{cases}x&\text{if }x\ge0\\-x&\text{if }x<0\end{cases}$

So we have two possible cases:

1) If $\dfrac{3x+1}{2x-7}\ge0$, then $\left|\dfrac{3x+1}{2x-7}\right|=\dfrac{3x+1}{2x-7}$, and solving the equation gives

$\dfrac{3x+1}{2x-7}=1\implies3x+1=2x-7\implies x=-8$

2) If $\dfrac{3x+1}{2x-7}<0$, then $\left|\dfrac{3x+1}{2x-7}\right|=-\dfrac{3x+1}{2x-7}$, and

$-\dfrac{3x+1}{2x-7}=1\implies\dfrac{3x+1}{2x-7}=-1\implies3x+1=-2x+7\implies5x=6$

$\implies x=\dfrac65$